Permutations and Combinations

C(45,9) = 886,163,135  different teams are possible

On a hockey team of 45 players, only 9 play at any given time.

How many different groups of people could be on the ice?

formula is:      nCr = 𝑛!/(𝑛−𝑟)!𝑟!

\(\begin{array}{|rcll|} \hline && nCr = \frac{n!}{(n-r)!r!} \qquad n= 45,\ r= 9 \\ \hline && \frac{45!}{(45-9)!9!} \\\\ &=& \frac{45!}{36!9!} \\\\ &=& \frac{36!\cdot 37\cdot 38\cdot 39\cdot 40\cdot 41\cdot 42\cdot 43\cdot 44\cdot 45}{36!9!} \\\\ &=& \frac{37\cdot 38\cdot 39\cdot 40\cdot 41\cdot 42\cdot 43\cdot 44\cdot 45}{9!} \\\\ &=& \frac{37\cdot 38\cdot 39\cdot 40\cdot 41\cdot 42\cdot 43\cdot 44\cdot 45}{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1} \\\\ &=& 37\cdot \frac{38}{2}\cdot \frac{39}{3}\cdot \frac{40}{8}\cdot 41\cdot \frac{42}{6\cdot 7}\cdot 43\cdot \frac{44}{4}\cdot \frac{45}{5\cdot 9} \\\\ &=& 37\cdot 19\cdot 13\cdot 5\cdot 41\cdot \frac{42}{42}\cdot 43\cdot 11\cdot \frac{45}{45} \\\\ &=& 37\cdot 19\cdot 13\cdot 5\cdot 41\cdot 43\cdot 11 \\\\ &=& \mathbf{886163135} \\ \hline \end{array}\)